{"paper":{"title":"Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"Korn\\'elia H\\'era","submitted_at":"2018-09-12T20:35:46Z","abstract_excerpt":"We show that if $B \\subset \\mathbb{R}^n$ and $E \\subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\\mathbb{R}^n$ such that every $P \\in E$ intersects $B$ in a set of Hausdorff dimension at least $\\alpha$ with $k-1 < \\alpha \\leq k$, then $\\dim B \\geq \\alpha +\\dim E/(k+1)$, where $\\dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $\\alpha$-Furstenberg set in the plane has Hausdorff dimension at least $\\alpha + 1/2$.\n  More generally, we prove that if $B$ and $E$ are as above with $0 < \\alpha \\leq k$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04666","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}